Assume a crosswell experiment
with one source and one receiver, and an interwell model
parameterization of one cell as shown
below.
For a single ray that passes
through the single slowness cell of width l,
the direct wave traveltime error dt
is related to the model error ds by

ds

=

dt/l.

(2.17)

In this case the error diminishes as l
becomes large, i.e., the longer we sample the
cell the smaller our slowness error for a given
traveltime error. Repeating the same experiment
N times will lead to different trav
eltime picking errors with the same raypath
the same raypath but
different slowness errors. Assuming a zero-mean traveltime error will
allow for an estimate of the slowness variance as:

[ ds2 ]

=

[ dt2 ]/ [ N l2 ] ,

(2.18)

where the brackets indicate an averaging over the different
outcomes (or random variables) of the experiment.
This equation says
that slowness variance is inversely proportional to the squared
segment length of the ray that passes through the cell times
the number of rays that pass through the cell.
Thus the model error can be assessed as a function of data errors and raypath segment
lengths. For arbitrary source-receiver
distributions and model parameterizations, the
covariance matrix
[LTL]-1
can be used to determine model errors in terms of the data errors.